From: Robin Chapman Subject: Re: approximation for g_2(L),g_3(L) (L=Lattice) Date: Thu, 26 Aug 1999 12:33:44 GMT Newsgroups: sci.math In article <37C514E4.AC14B094@stud.uni-hannover.de>, Stephan Bauer wrote: > Hi. > > I want to calculate g_2(L) and g_3(L) where L (subset of C) is a > lattice and > g_2= 60 * sum_{w in L\{0}} 1/w^4 > g_3= 140 * sum_{w in L\{0}} 1/w^6. > I am looking for a numerical approximation for g_2 and g_3 > (references). These are standard examples of modular forms. Let's assume that L has generators 1 and w where Im(w) > 0. [The general case comes easily from this.] In Serre's "A course in arithmetic" one finds the formulas g_2 = (4 pi^4/3) E_2 and g_3 = (8 pi^6/27) E_3 where E_2 = 1 + 240 sum_{n=1}^infinity sigma_3(n) q^n and E_3 = 1 - 504 sum_{n=1}^infinity sigma_5(n) q^n. Here q = exp(2 pi i w) and sigma_j(n) is the sum of the j-th powers of the positive integer divisors of n. One can ensure that Im(w) >= 1/2 and so the convergence will be reasonably fast. -- Robin Chapman http://www.maths.ex.ac.uk/~rjc/rjc.html "They did not have proper palms at home in Exeter." Peter Carey, _Oscar and Lucinda_ Sent via Deja.com http://www.deja.com/ Share what you know. Learn what you don't.